The character of Mrs. Pumphrey and the episodes involving her pampered pet, Tricki Woo, in James Herriot's "All Creatures Great and Small" are based on Herriot's real-life experiences as a veterinarian. While some details and events are likely fictionalized or exaggerated for literary effect, the core of the story is rooted in Herriot's observations and encounters with real people and their pets. So, it can be considered a mixture of both real-life experiences and fictional elements.
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD. 
In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $\angle OPQ = 15^\circ$ and $\angle PTQ = \theta$, then find the value of $\sin 2\theta$. 